Backward Stochastic Differential Equations with no driving martingale, Markov processes and associated Pseudo Partial Differential Equations

0.45 MB·

by  Adrien Barrasso

Backward Stochastic Differential Equations with
7 no driving martingale, Markov processes and
1
0
associated Pseudo Partial Differential Equations.
2

r
a
Adrien BARRASSO ∗ Francesco RUSSO†
M

8
February 2017

]
R
P
Abstract. WeinvestigateexistenceanduniquenessforanewclassofBack-
.
h ward Stochastic Differential Equations (BSDEs) with no driving martingale.
t When the randomness of the driver depends on a general Markov process X,
a
m those BSDEs are denominated forward BSDEs and can be associated to a de-
terministic problem, called Pseudo-PDE which constitute the natural general-
[
izationofa parabolicsemilinearPDE whichnaturallyappearswhenthe under-

2
lying filtration is Brownian. We consider two aspects of well-posedness for the
v
Pseudo-PDEs: classical and martingale solutions.
9
9
8 MSC 2010 Classification. 60H30; 60H10; 35S05;60J35; 60J60;60J75.
2
0 KEY WORDS AND PHRASES. Martingale problem; pseudo-PDE;
.
1 Markov processes; backwardstochastic differential equation.
0
7
1 1 Introduction
:
v
i This paper focuses ona new conceptofBackwardStochastic DifferentialEqua-
X
tion (in short BSDE) with no driving martingale of the form
r
a
T d M
Y =ξ+ fˆ r, ,Y , h i(r) dV (M M ), (1.1)
t r r T t
Zt · r dV ! − −
defined on a fixed stochastic basis fulfilling the usual conditions. V is a given
non-decreasingcontinuousadaptedprocess, ξ (resp. fˆ) is a prescribedterminal
∗ENSTA ParisTech, Unité de Mathématiques appliquées, 828, boulevard des Maréchaux,
F-91120 Palaiseau, France and Ecole Polytechnique, F-91128 Palaiseau, France. E-mail:
[email protected]
†ENSTA ParisTech, Unité de Mathématiques appliquées, 828, boulevard des Maréchaux,
F-91120Palaiseau,France. E-mail: [email protected]
1
condition (resp. driver). The unknown will be a couple of cadlag adapted
processes(Y,M)whereM isamartingale. AparticularcaseofsuchBSDEsare
the forwardBSDEs (in short FBSDEs) of the form
T d Ms,x
Ys,x =g(X )+ f r,X ,Ys,x, h i(r) dV (Ms,x Ms,x), (1.2)
t T Zt r r r dV ! r− T − t
defined in a canonical space Ω, s,x,(X ) ,( s,x) , s,x
where ( s,x) correspFonds totthte∈[0la,Tw]s (Ffotr dti∈ff[e0r,Ten]tPstarting times s
P (s,x)∈[0,T]×E (cid:0) (cid:1)
andstartingpointsx)ofanunderlyingforwardMarkovprocesswithtimeindex
[0,T], taking values in a Polish state space E. Indeed this Markov process is
supposed to solve a martingale problem with respect to a given deterministic
operatora,whichisthenaturalgeneralizationofstochasticdifferentialequation
inlaw. (1.2)willbenaturallyassociatedwithadeterministicprobleminvolving
a, which will be called Pseudo-PDE, being of the type
a(u)(t,x)+f t,x,u(t,x), Γ(u,u)(t,x) = 0 on [0,T] E
× (1.3)
( (cid:16) p u(T,(cid:17)) = g,
·
where Γ(u,u) = a(u2) 2ua(u) is a potential theory operator called the carré
−
du champs operator. The forward BSDE (1.2) seems to be appropriated in the
case when the forwardunderlying process X is a generalMarkovprocesswhich
does notrely to a fixedreference processorrandomfield as a Brownianmotion
or a Poisson measure.
The classical notion of Brownian BSDE was introduced in 1990 by E. Par-
douxandS.Pengin[26],afteranearlyworkofJ.M.Bismutin1973in[9]. Itisa
stochastic differential equation with prescribed terminal condition ξ and driver
fˆ; the unknown is a couple (Y,Z) of adapted processes. Of particular interest
is the case when the randomness of the driver is expressed through a forward
diffusion process X and the terminal condition only depends on X . The solu-
T
tion,whenitexists,is usuallyindexedbythe startingtime s andstartingpoint
x of the forward diffusion X =Xs,x, and it is expressed by
Xs,x = x+ tµ(r,Xs,x)dr+ tσ(r,Xs,x)dB
t s r s r r (1.4)
( Yts,x = g(XTRs,x)+ tT f(r,Xrs,Rx,Yrs,x,Zrs,x)dr− tT Zrs,xdBr,
where B is a Brownian moRtion. Existence and uniquenRess of (1.4) (that we
still indicate with FBSDE) above was established first supposing essentially
Lipschitz conditions on f with respect to the third and fourth variable. µ and
σ were also supposed to be Lipschitz (with respect to x). In the sequel those
conditions were considerably relaxed, see [28] and references therein.
In [29] and in [27] previous FBSDE was linked to the semilinear PDE
∂ u+ 1 (σσ⊺) ∂2 u+ µ ∂ u+f((, ),u,σ u)=0 on [0,T[ d
t 2 i,j xixj i xi · · ∇ ×R
i,j≤d i≤d
( u(T, )=Pg. P
·
(1.5)
2
In particular, if (1.5) has a classical smooth solution u then (Ys,x,Zs,x) :=
(u(,Xs,x),σ u(,Xs,x)) solves the second line of (1.4). Conversely, only un-
· ·
· ∇ ·
der the Lipschitz type conditions mentioned after (1.4), the solution of the
FBSDE can be expressed as a function of the forward process (Ys,x,Zs,x) =
(u(,Xs,x),v(,Xs,x)), see [17]. When f and g are continuous, u is a viscosity
· ·
· ·
solutionof (1.5). Excepted in the case when u has some minimal differentiabil-
ity properties, see e.g. [19], it is difficult to say something more on v.
Sincethepioneeringworkof[27],intheBrowniancase,therelationsbetween
more general BSDEs and associated deterministic problems have been studied
extensively, and innovations have been made in several directions.
In[5]theauthorsintroducedanewkindofFBSDEincludingatermwithjumps
generatedby a Poissonmeasure,where anunderlying forwardprocessX solves
ajump diffusion equationwithLipschitz type conditions. They associatedwith
it anIntegral-PartialDifferential Equation(in shortIPDE) in which some non-
local operators are added to the classical partial differential maps, and proved
that, under some continuity conditions on the coefficients, the BSDE provides
a viscosity solution of the IPDE. In chapter 13 of [6], under some specific con-
ditions on the coefficients of a Brownian BSDE, one produces a solution in the
sense of distributions of the parabolic PDE. Later, the notion of mild solution
ofthe PDE wasusedin [3]where the authorstackleddiffusionoperatorsgener-
atingsymmetricDirichletformsandassociatedMarkovprocessesthanksto the
theory of Fukushima Dirichlet forms, see e.g. [20]. Infinite dimensional setups
were considered for example in [19] where an infinite dimensional BSDE could
produce the mild solution of a PDE on a Hilbert space. Concerning the study
of BSDEs driven by more general martingales than Brownian motion, we have
already mentioned BSDEs driven by Poisson measures. In this respect, more
recently, BSDEs driven by marked point processes were introduced in [12], see
also[4];inthatcasetheunderlyingprocessdoesnotcontainanydiffusionterm.
Brownian BSDEs involving a supplementary orthogonal term were studied in
[17]. We can also mention the study of BSDEs driven by a general martin-
gale in [10]. BSDEs of the same type, but with partial information have been
investigated in [11]. A first approach to face deterministic problems for those
equations appears in [24]; that paper also contains an application to financial
hedging in incomplete market.
We come back to the motivations of the paper. Besides introducing and
studying the new class of BSDEs (1.1), (resp. forward BSDEs (1.2)), we study
thecorrespondingPseudo-PDE(1.3)andcarefullyexploretheirrelationsinthe
spiritoftheexistinglinksbetween(1.4)and(1.5). ForthePseudo-PDE,wean-
alyzewell-posednessattwo differentlevels: classical solutions,which generalize
the C1,2-solutions of (1.5) and the so called martingale solutions. In a paper
in preparation [7], we will also discuss other (analytical) solutions, that we de-
nominateasmild solutions. The maincontributionsofthe paperareessentially
the following. In Section 3 we introduce the notion of BSDE with no driving
martingale (1.1). Theorem 3.22 states existence and uniqueness of a solution
3
for that BSDE, when the final condition ξ is square integrable and the driver
fˆverifies some integrabilityand Lipschitz conditions. In Section 4, we consider
an operator and its domain (a, (a)); V will be a continuous non-decreasing
D
function. That section is devoted to the formulation of the martingale problem
concerning our underlying process X. For each initial time s and initial point
x the solution will be a probability s,x under which for any φ (a),
P ∈D
·
φ(,X) φ(s,x) a(φ)(r,X )dV
· r r
· − −
Zs
is a local martingale starting in zero at time s. We will then assume that
this martingale problem is well-posed and that its solution ( s,x)
(s,x)∈[0,T]×E
P
definesaMarkovprocess. InProposition4.11,weprovethat, undereachoneof
those probabilities,the angularbracketof everysquare integrablemartingaleis
absolutelycontinuouswithrespecttodV. InDefinition4.15,wesuitably define
some extended domains for the operators a and Γ, using some locally convex
topology. InSection5weintroducethePseudo-PDE(1.3)towhichweassociate
the FBSDE (1.2), considered under every s,x. We also introduce the notions
P
of classical solution in Definition 5.1, and of martingale solution in Definition
5.17, which is fully probabilistic. Classical solutions of (1.3) typically belong
to the domain (a) and are shown also to be essentially martingale solutions,
D
see Proposition 5.19. Conversely a martingale solution belonging to (a) is a
D
classical solution, up to so called zero potential sets, see Definition 4.12. In
Theorem5.14, we showthat, without anyassumptions ofregularity,there exist
Borel functions u and v such that for any (s,x) [0,T] E, the solution of
∈ ×
(1.2) verifies
t s:Ys,x =u(t,X ) s,x a.s.
∀dhM≥s,xi(t)t=v2(t,X )t dVP d s,x a.e.
(cid:26) dV t ⊗ P
Theorems 5.20 and 5.21 state that the function u is the unique martingale
solution of (1.3). Proposition 5.8 asserts that, given a classical solution u
∈
(a), then for any (s,x) the processes Ys,x =u(,X) and
·
DMs,x =u(,X) u(s,x) ·f((, ,u, Γ(u,u))(r·,X )dV solve(1.2)underthe
· · − − s · · r r
probability s,x. In Section 6 we list some examples which will be developed
P R p
in [7]. These include Markov processes defined as weak solutions of Stochastic
Differential Equations (in short SDEs) including possible jump terms, α-stable
LévyprocessesassociatedtofractionalLaplaceoperators,andsolutionsofSDEs
with distributional drift.
2 Preliminaries
In the whole paper we will use the following notions,notations and vocabulary.
A topological space E will always be considered as a measurable space with
its Borel σ-field which shall be denoted (E) and if (F,d ) is a metric space,
F
B
(E,F) (respectively (E,F), (E,F), (E,F)) will denote the set of func-
b b
C C B B
tionsfromEtoF whicharecontinuous(respectivelyboundedcontinuous,Borel,
4
bounded Borel).
On a fixed probability space (Ω, , ), for any p 1, Lp will denote the set
F P ≥
of random variables with finite p-th moment. A measurable space equipped
with a right-continuous filtration (Ω, ,( ) ) (where is equal to or
t t∈ +
to [0,T] for some T ∗) will beFcalFled aTfiltered sTpace. A proRbabil-
∈ R+
ity space equipped with a right-continuous filtration (Ω, ,( ) , ) will be
t t∈
called called a stochastic basis and will be said to fulfiFll tFhe uTsuPal condi-
tionsiftheprobabilityspaceiscompleteandif containsallthe -negligible
0
F P
sets. Weintroducenowsomenotationsandvocabularyaboutspacesofstochas-
tic processes, on a fixed stochastic basis (Ω, ,( ) , ). Most of them are
t t∈
taken or adapted from [22] or [23]. A processF(XF) TisPsaid to be integrable
t t∈
if X is an integrable r.v. for any t. We will denoTte (resp +) the set of
t
V V
adapted, bounded variation (resp non-decreasing) processes starting at 0; p
V
(resp p,+) the elements of (resp +) which are predictable, and c (resp
V V V V
c,+) the elements of (resp +) which are continuous; will be the space
V V V M
of cadlag martingales. For any p [1, ] p will denote the Banach space
of elements of for which M Hp∈:= ∞[suHp Mt p]p1 < and in this set we
M k k E|t∈ | ∞
identify indistinguishable elements. p will dTenote the Banach subspace of p
H0 H
of elements vanishing at zero.
If =[0,T]forsomeT ∗,astoppingtimewilltakevaluesin[0,T] + .
T ∈R+ ∪{ ∞}
Wedefinealocalizingsequenceofstoppingtimesasanincreasingsequence
of stopping times (τ ) such that there exists N for which τ = + .
n n≥0 N
Let Y be a process and τ a stopping time, we denote∈byNYτ the stopped pr∞o-
cess t Y . If is a set of processes, we define its localized class as
t∧τ loc
7→ C C
the set of processes Y such that there exist a localizing sequence (τ ) such
n n≥0
that for every n, the stopped process Yτn belongs to . In particular a process
C
X is said to be locally integrable (resp. locally square integrable) if there is
a localizing sequence (τ ) such that for every n, Xτn is integrable (resp.
n n≥0 t
square integrable) for every t.
For any M , we denote [M] its quadratic variation and if moreover
loc
∈ M
M 2 , M will denote its (predictable) angular bracket. 2 will be
∈ Hloc h i H0
equipped with scalar product defined by (M,N) = [M N ]= [ M,N ]
H20 E T T Eh iT
which makes it a Hilbert space. Two local martingales M,N will be said to be
strongly orthogonal if MN is a local martingale starting in 0 at time 0. In
2 this notion is equivalent to M,N =0.
H0,loc h i
3 BSDEs without driving martingale
In the whole present section we are given T ∗, and a stochastic basis
∈ R+
Ω, ,( ) , fulfilling the usual conditions. Some proofs and interme-
t t∈[0,T]
F F P
diary results of the first part of this section are postponed to Appendix B.
(cid:0) (cid:1)
Definition 3.1. Let A and B be in +. We will say that dB dominates dA in
V
the sense of stochastic measures (written dA dB) if for almost all ω,
≪
5
dA(ω) dB(ω) as Borel measures on [0,T].
≪
We will say that dB and dA are mutually singular in the sense of stochastic
measures (written dA dB) if for almost all ω, the Borel measures dA(ω) and
⊥
dB(ω) are mutually singular.
Let B +. dB d will denote the positive measure on
∈V ⊗ P
(Ω [0,T], ([0,T])) defined for any F ([0,T]) by
× F ⊗B ∈F ⊗B
dB d (F)= T 1 (r,ω)dB (ω) . A property which holds true everywhere
⊗ P E 0 F r
except on a null hsRet for this measureiwill be said to be true dB ⊗dP almost
everywhere (a.e).
The proof of Proposition below is in Appendix B.
Proposition 3.2. For any A and B in p,+, there exists a (non-negative
dB d a.e.) predictable process dA andVa process in p,+ A⊥B such that
⊗ P dB V
dA⊥B dB and A=AB +A⊥B a.s.
⊥
where AB = · dA(r)dB . The process A⊥B is unique and the process dA is
0 dB r dB
unique dB d a.e.
⊗ RP
Moreover, there exists a predictable process K with values in [0,1] (for every
(ω,t)), such that AB = ·1 dA and A⊥B = ·1 dA .
0 {Kr<1} r 0 {Kr=1} r
The predictable procRess dA appearing in the stRatement of Proposition 3.2
dB
will be called the Radon-Nikodym derivative of A by B.
Remark 3.3. Since for any s < t A A = t dA(r)dB +A⊥B A⊥B a.s.
t− s s dB r t − s
where A⊥B is increasing, it is clear that for any s<t,
t dA(r)dB A A a.s. and therefore thRat for any positive measurable
s dB r ≤ t − s
process φ we have T φ dA(r)dB T φ dA a.s.
R 0 rdB r ≤ 0 r r
If A is in , weRwill denote A+ anRd A− the positive variation and negative
V
variation parts of A, meaning the unique pair of elements + such that A =
V
A+ A−, see Proposition I.3.3 in [23] for their existence. If A is in p, and
B −p,+. We set dA := dA+ dA− and A⊥B :=(A+)⊥B (A−)⊥B. V
∈V dB dB − dB −
Proposition 3.4. Let A and A be in p, and B p,+. Then,
1 2
V ∈V
d(A1+A2) = dA1 + dA2 dV d a.e. and (A +A )⊥B =A⊥B +A⊥B.
dB dB dB ⊗ P 1 2 1 2
Proof. The proof is an immediate consequence of the uniqueness of the decom-
position (3.2).
Let V p,+. We introduce two significant spaces related to V.
∈V
2,V := M 2 d M dV and 2,⊥V := M 2 d M dV .
H { ∈H0| h i≪ } H { ∈H0| h i⊥ }
The proof of Proposition below is in Appendix B.
6
Proposition 3.5. Let M 2, and let V p,+. There exists a pair
∈ H0 ∈ V
(MV,M⊥V)in 2,V 2,⊥V suchthatM =MV+M⊥V and MV,M⊥V =0.
H ×H h i
Moreover, we have MV = M V = · dhMi(r)dV and M⊥V = M ⊥V
h i h i 0 dV r h i h i
and there exists a predictable process K with values in [0,1] such that
MV = ·1 dM and M⊥V = ·1 R dM .
0 {Kr<1} r 0 {Kr=1} r
TheRproof of the proposition beloRw is in Appendix B.
Proposition3.6. 2,V and 2,⊥V areorthogonal sub-Hilbertspaces of 2 and
H H H0
2 = 2,V ⊥ 2,⊥V. Moreover, any element of 2,V is strongly orthogonal to
H0 H ⊕ H Hloc
any element of 2,⊥V.
Hloc
Remark 3.7. All previous results extend when the filtration is indexed by .
+
R
We are going to introduce here a new type of Backward Stochastic Dif-
ferential Equation (BSDE) for which there is no need for having a particular
martingale of reference.
We willdenote rothe σ-fieldgeneratedby progressivelymeasurableprocesses
P
defined on [0,T] Ω.
×
Given some V c,+, we will indicate by 2(dV d ) (resp. 0(dV d ))
∈ V L ⊗ P L ⊗ P
the setof(upto indistinguishability)progressivelymeasurableprocessesφsuch
that [ T φ2dV ] < (resp. T φ dV < a.s.) and L2(dV d ) the
E 0 r r ∞ 0 | r| r ∞ P ⊗ P
quotient space of 2(dV d ) with respect to the subspace of processes equal
R L ⊗ P R
to zero dV d a.e. More formally, L2(dV d ) corresponds to the classical
⊗ P ⊗ P
L2 space L2([0,T] Ω, ro,dV d ) and is therefore complete for its usual
× P ⊗ P
norm.
2,cadlag(dV d ) (resp. L2,cadlag(dV d )) will denote the subspace of
L ⊗ P ⊗ P
2(dV d ) (resp. L2(dV d )) of cadlagelements (resp. of elements having
aLcadla⊗g rePpresentative). W⊗e emPphasize that L2,cadlag(dV d ) is not a closed
⊗ P
subspace of L2(dV d ).
The application wh⊗ichPto a process associate its class will be denoted φ φ˙.
7→
We will now consider some bounded V c,+, an -measurablerandomvari-
T
able ξ calledthe final conditionanda∈dVriver fˆ:F([0,T] Ω) ,
measurable with respect to ro ( ) ( ). We wi×ll ass×umRe×thRat−→(ξ,Rfˆ)
P ⊗B R ⊗B R
verify the following hypothesis.
Hypothesis 3.8.
1. ξ L2;
∈
2. fˆ(, ,0,0) 2(dV d );
· · ∈L ⊗ P
3. There exist positive constants KY,KZ such that, a.s. we have for all
P
t,y,y′,z,z′,
fˆ(t, ,y,z) fˆ(t, ,y′,z′) KY y y′ +KZ z z′ . (3.1)
| · − · |≤ | − | | − |
7
We start with a lemma.
Lemma 3.9. Let U and U be in 2(dV d )and such that U˙ =U˙ . Let F :
1 2 1 2
L ⊗ P
[0,T] Ω besuchthatF((, ),U )andF(, ,U )arein 0(dV d ),
1 2
then t×hep×roRce−ss→esR·F(r,ω,U1)dV a·n·d ·F(r,ω,U·2)·dV areindisLtinguis⊗habPle.
0 r r 0 r r
Proof. There existRs a -null set suchRthat for any ω c, U1(ω) = U2(ω)
dV(ω) a.e. So for anyPω c, FN(,ω,U1(ω)) = F(,ω,U∈2N(ω)) dV(ω) a.e. im-
plying ·F(r,ω,U1(ω))dV∈(Nω) = ··F(r,ω,U2(ω))d·V (ω). So ·F(r, ,U1)dV
and ·F0(r, ,U2)dVr are inrdistingu0ishable prorcesses. r 0 · r r
0R · r r R R
InR some of the following proofs, we will have to work with classes of pro-
cesses. AccordingtoLemma3.9,ifU˙ isanelementofL2(dV d ),theintegral
· ⊗ P
F(r,ω,U )dV willnotdependontherepresentantiveprocessU thatwehave
0 r r
chosen.
R
We will now give the formulation of our BSDE.
Definition 3.10. We say that a couple (Y,M) 2,cadlag(dV d ) 2 is a
solution of BSDE(ξ,fˆ,V) if it verifies ∈L ⊗ P ×H0
T d M
Y =ξ+ fˆ r, ,Y , h i(r) dV (M M) (3.2)
r r T ·
Z· · r dV ! − −
in the sense of indistinguishability.
Proposition 3.11. If (Y,M) solves BSDE(ξ,fˆ,V), and if we denote
fˆ r, ,Y , dhMi(r) by fˆ, then for any t [0,T], a.s. we have
· r dV r ∈
(cid:18) q (cid:19)
Y = ξ+ T fˆdV
t E t r r Ft (3.3)
 Mt = Ehξ+R0T fˆrdVr(cid:12)(cid:12)(cid:12)Fti−E ξ+ 0T fˆrdVr F0 .
Proof. Since Yt =ξ+ tThfˆrdVRr −(MT(cid:12)(cid:12)(cid:12)−Mit) a.sh., Y Rbeing an(cid:12)(cid:12)(cid:12)adaipted process
and M a martingale, taking the expectation in (3.2) at time t, we directly get
R
Y = ξ+ T fˆdV and in particular that Y = ξ+ T fˆdV .
t E t r r Ft 0 E 0 r r F0
Sξi+nce0TMfˆr0hd=Vr0−R,lEookξin+g(cid:12)(cid:12)(cid:12)a0TttifˆhredVBrSFD0E.atTtaikmineg0twheeegxepteMctTat=ionξh+wiRt0hTRrfˆersdpVerc−tt(cid:12)(cid:12)(cid:12)Yo0F=it
in the above ineqhuality gives th(cid:12)e seicond line of (3.3).
R R (cid:12)
(cid:12)
We willproceedshowingthatBSDE(ξ,fˆ,V)hasaunique solution. Atthis
point we introduce a significant map Φ which will map L2(dV d ) 2 into
its subspaceL2,cadlag(dV d ) 2. Fromnowon, untilNota⊗tionP3.×18H,w0e fix
a couple (U˙,N) L2(dV ⊗dP)×H02 to which we will associate (Y˙,M) which,
as we will show,∈will belo⊗ngPto×L2H,c0adlag(dV d ) 2. We will show that
(U˙,N) (Y˙,M) is a contraction for a certai⊗n nPorm×. HIn0all the proofs below,
U˙ will o7→nly appear in integrals driven by dV through a representative U.
8
Proposition 3.12. For any t [0,T], T fˆ2 r, ,U , dhNi(r) dV is in L1
∈ t · r dV r
(cid:18) q (cid:19)
and T fˆ r, ,U , dhNi(r) dV isRin L2.
t · r dV r
(cid:18) (cid:18) q (cid:19) (cid:19)
R
Proof. By Jensen’s inequality and thanks to the Lipschitz conditions on f in
Hypothesis 3.8, there exist a positive constant C such that, for any t [0,T],
∈
we have
2
T fˆ r, ,U , dhNi(r) dV T fˆ2 r, ,U , dhNi(r) dV
t · r dV r ≤ t · r dV r (3.4)
(cid:18)RC (cid:18)T fˆ2(r, ,q0,0)dV +(cid:19) T U(cid:19)2dVR+ T(cid:18)dhNi(r)dqV . (cid:19)
≤ t · r t r r t dV r
The thre(cid:16)eRterms on the right arRe in L1. IndReed, by Remar(cid:17)k 3.3
T dhNi(r)dV ( N N ) which belongs to L1 since N is taken in 2.
t dV r ≤ h iT −h it H
By Hypothesis 3.8, f(, ,0,0) is in 2(dV d ), and U˙ was also taken in
R · · L ⊗ P
L2(dV d ). This concludes the proof.
⊗ P
We can therefore state the following definition.
Definition 3.13. Setting fˆ = fˆ r, ,U , dhNi(r) , l ttet M be the cadlag
r · r dV
(cid:18) q (cid:19)
version of the martingale t ξ+ T fˆdV ξ+ T fˆdV .
7→E 0 r r Ft −E 0 r r F0
M is square integrable by ProhpositRion 3.12.(cid:12)(cid:12) Itiadmiths a cRadlag ver(cid:12)(cid:12)sioni tak-
ing into account Theorem 4 in Chapter IV of(cid:12) [14], since the stocha(cid:12)stic basis
fulfills the usual conditions. We denote by Y the cadlag process defined by
Y = ξ + T fˆ r, ,U , dhNi(r) dV (M M ). This will be called the
t t · r dV r − T − t
cadlag reRferen(cid:18)ce procqess and w(cid:19)e will often omit its dependence to (U˙,N).
According to previous definition, it is not clear whether Y is adapted, how-
ever, we have the almost sure equalities
Y = ξ+ T fˆdV (M M )
t t r r − T − t
= ξ+ T fˆdV ξ+ T fˆdV ξ+ T fˆdV
Rt r r − 0 r r −E 0 r r Ft
(3.5)
= E ξR+ 0T fˆrdVr(cid:16)Ft −R 0tfˆrdVr h R (cid:12)(cid:12)(cid:12) i(cid:17)
= Ehξ+RtT fˆrdVr(cid:12)(cid:12)(cid:12)Fti. R
h (cid:12) i
R (cid:12)
Since Y iscadlagandadapted,(cid:12)byTheorem15ChapterIVof[13],itis progres-
sively measurable.
Proposition 3.14. Y and M are square integrable processes.
Proof. We already know that M is a square integrable martingale. As we have
seen in Proposition 3.12, T fˆ r, ,U , dhNi(r) dV belongs to L2 for any
t · r dV r
(cid:18) q (cid:19)
R
9
t [0,T] and by Hypothesis 3.8, ξ L2. So by (3.5) and Jensen’s inequality
∈ ∈
for conditional expectation we have
2
Y2 = ξ+ T fˆ r, ,U , dhNi(r) dV
E t E"E(cid:20) t (cid:18) · r q dV (cid:19) r(cid:12)Ft(cid:21) #
(cid:2) (cid:3) R (cid:12) 2
≤ E"E"(cid:18)ξ+ tT fˆ(cid:18)r,·,Ur,qddhNVi(r)(cid:19)dV(cid:12)(cid:12)r(cid:19) (cid:12)(cid:12)Ft##
R 2 (cid:12)
= ξ+ T fˆ r, ,U , dhNi(r) dV , (cid:12)
E"(cid:18) t (cid:18) · r q dV (cid:19) r(cid:19) # (cid:12)
R
which is finite.
Lemma 3.15. Let Y be a cadlag adapted process satisfying sup Y2 <
E"t∈[0,T] t # ∞
and M be a square integrable martingale. Then there exists a constant C > 0
such that for any ǫ>0 we have
t ǫ 1
E"t∈su[0p,T](cid:12)Z0 Yr−dMr(cid:12)#≤C 2E"t∈su[0p,T]Yt2#+ 2ǫE[[M]T]!.
(cid:12) (cid:12)
(cid:12) (cid:12)
In particular, 0·Y(cid:12)r−dMr is a(cid:12)uniformly integrable martingale.
Proof. By BurRkholder-Davis-Gundy (shortened by BDG) and Cauchy-Schwarz
(shortened by CS) inequalities, there exists C >0 such that
E"t∈su[0p,T](cid:12) 0tYr−dMr(cid:12)# ≤ CE(cid:20)q 0T Yr2−d[M]r(cid:21)
(cid:12)R (cid:12) R
(cid:12) (cid:12)
C sup Y2[M] C sup Y2 [[M]]
≤ E" t∈[0,T] t T# ≤ vuE"t∈[0,T] t #E T
r u
t
C ǫ sup Y2 + 1 [[M] ] < + .
≤ 2E"t∈[0,T] t # 2ǫE T ! ∞
·
So 0Yr−dMr is a uniformly integrable local martingale, and therefore a mar-
tingale.
R
Lemma 3.16. Let Y be a cadlag adapted process and M 2. Assume the
∈ H
existence of a constant C > 0 and an L1 random variable Z such that for any
t∈[0,T], Yt2 ≤C Z+ 0tYr−dMr . Then sup |Yt|∈L2.
t∈[0,T]
(cid:16) (cid:12)(cid:12)R (cid:12)(cid:12)(cid:17)
Proof. For any stopping(cid:12)time τ we h(cid:12)ave
t
ts∈u[0p,τ]Yt2 ≤C Z+ts∈u[0p,τ](cid:12)Z0 Yr−dMr(cid:12)!. (3.6)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
10